Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1

Average Error: 14.4 → 1.2

Time: 28.0s

Precision: 64

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.5398666775420435 \cdot 10^{+227}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -8.248772990819591 \cdot 10^{-297}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(y \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r2334010 = x;
        double r2334011 = y;
        double r2334012 = z;
        double r2334013 = r2334011 / r2334012;
        double r2334014 = t;
        double r2334015 = r2334013 * r2334014;
        double r2334016 = r2334015 / r2334014;
        double r2334017 = r2334010 * r2334016;
        return r2334017;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r2334018 = y;
        double r2334019 = z;
        double r2334020 = r2334018 / r2334019;
        double r2334021 = -2.5398666775420435e+227;
        bool r2334022 = r2334020 <= r2334021;
        double r2334023 = x;
        double r2334024 = r2334023 * r2334018;
        double r2334025 = r2334024 / r2334019;
        double r2334026 = -8.248772990819591e-297;
        bool r2334027 = r2334020 <= r2334026;
        double r2334028 = r2334020 * r2334023;
        double r2334029 = cbrt(r2334023);
        double r2334030 = r2334029 * r2334029;
        double r2334031 = cbrt(r2334019);
        double r2334032 = r2334031 * r2334031;
        double r2334033 = r2334030 / r2334032;
        double r2334034 = r2334029 / r2334031;
        double r2334035 = r2334018 * r2334034;
        double r2334036 = r2334033 * r2334035;
        double r2334037 = r2334027 ? r2334028 : r2334036;
        double r2334038 = r2334022 ? r2334025 : r2334037;
        return r2334038;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -2.5398666775420435e+227

   1. Initial program 42.6

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 0.6

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 1.8

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \cdot y\]

   5. Applied add-cube-cbrt 2.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \cdot y\]

   6. Applied times-frac 2.0

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)} \cdot y\]

   7. Applied associate-*l* 1.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot y\right)}\]
   8. Using strategy rm

   9. Applied associate-*l/ 1.8

      \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \color{blue}{\frac{\sqrt[3]{x} \cdot y}{\sqrt[3]{z}}}\]

   10. Applied frac-times 2.3

       \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot y\right)}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   11. Simplified 2.1

       \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   12. Simplified 0.8

       \[\leadsto \frac{x \cdot y}{\color{blue}{z}}\]

   if -2.5398666775420435e+227 < (/ y z) < -8.248772990819591e-297

   1. Initial program 9.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 8.3

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y}\]
   3. Using strategy rm

   4. Applied div-inv 8.4

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{z}\right)} \cdot y\]

   5. Applied associate-*l* 0.3

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{z} \cdot y\right)}\]

   6. Simplified 0.2

      \[\leadsto x \cdot \color{blue}{\frac{y}{z}}\]

   if -8.248772990819591e-297 < (/ y z)

   1. Initial program 15.0

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 5.6

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 6.3

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \cdot y\]

   5. Applied add-cube-cbrt 6.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \cdot y\]

   6. Applied times-frac 6.5

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)} \cdot y\]

   7. Applied associate-*l* 1.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot y\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.5398666775420435 \cdot 10^{+227}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -8.248772990819591 \cdot 10^{-297}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(y \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))